Orbital stability in the cubic defocusing NLS equation: I. Cnoidal periodic waves

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ScienceDirect J. Differential Equations 258 (2015) 3607–3638 www.elsevier.com/locate/jde

Orbital stability in the cubic defocusing NLS equation: I. Cnoidal periodic waves Thierry Gallay a , Dmitry Pelinovsky b,∗ a Institut Fourier, Université de Grenoble 1, 38402 Saint-Martin-d’Hères, France b Department of Mathematics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

Available online 9 February 2015

Abstract Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in [4] to be linearly stable and the Floquet–Bloch spectrum of the linearized operator has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that cnoidal periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. Our result is not restricted to the periodic waves of small amplitudes. © 2015 Elsevier Inc. All rights reserved.

1. Introduction We consider the cubic defocusing NLS (nonlinear Schrödinger) equation in one space dimension: iψt + ψxx − |ψ|2 ψ = 0,

(1.1)

where ψ = ψ(x, t) ∈ C and (x, t) ∈ R × R. This equation arises in the study of modulational stability of small amplitude nearly harmonic waves in nonlinear dispersive systems [14]. In this context, monochromatic waves of the original system correspond to spatially homogeneous 2 solutions of the cubic NLS equation (1.1) of the form ψ(x, t) = ae−ia t , where the positive * Corresponding author.

E-mail address: [email protected] (D. Pelinovsky). http://dx.doi.org/10.1016/j.jde.2015.01.018 0022-0396/© 2015 Elsevier Inc. All rights reserved.

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parameter a can be taken equal to one without loss of generality, due to scaling invariance. According to the famous Lighthill criterion, these plane waves are spectrally stable with respect to sideband perturbations [16], because the nonlinearity in (1.1) is defocusing. Moreover, using energy methods, it can be shown that plane waves are also orbitally stable under perturbations in H 1 (R) [17, Section 3.3], where the orbit is defined with respect to arbitrary rotations of the complex phase of ψ . More generally, it is important for the applications to consider spatially inhomogeneous waves of the form ψ(x, t) = u0 (x)e−it , where the profile u0 : R → C satisfies the second-order differential equation  d 2 u0  + 1 − |u0 |2 u0 = 0, 2 dx

x ∈ R.

(1.2)

Such solutions of the cubic NLS equation (1.1) correspond to slowly modulated wave trains of the original physical system. A complete list of all bounded solutions of the second-order equation (1.2) is known, see [4,7]. Most of them are quasi-periodic in the sense that u0 (x) = r(x)eiϕ(x) for some real-valued functions r, ϕ such that r and ϕ  are periodic with the same period T0 > 0. The corresponding solutions of the cubic NLS equation (1.1) are usually called “periodic waves”, although strictly speaking they are not periodic functions of x in general. In addition, the second-order equation (1.2) has nonperiodic solutions such that r and ϕ  converge to a limit as x → ±∞; these correspond to “dark solitons” of the cubic NLS equation. In the present paper, we focus on real-valued solutions of the second-order equation (1.2), which form a one-parameter family of periodic waves (often referred to as “cnoidal waves”). Several recent works addressed the stability of periodic waves for the cubic NLS equation (1.1). Using the energy method, it was shown in [6,7] that periodic waves are orbitally stable within a class of solutions which have the same periodicity properties as the wave itself. More precisely, if u0 (x) = eipx q0 (x) where p ∈ R and q0 is T0 -periodic, the wave u0 (x)e−it is orbitally 1 (0, T ). Here stable among solutions of the form ψ(x, t) = ei(px−t) q(x, t), where q(·, t) ∈ Hper 0 the orbit is defined with respect to translations in space and rotations of the complex phase. The proof follows the general strategy proposed in [8] and relies on the fact that the periodic wave is a constrained minimizer of the energy   E(ψ) =

|ψx |2 + I

 2 1 1 − |ψ|2 dx, 2

(1.3)

subject to fixed values of the charge Q and the momentum M given by  Q(ψ) =

|ψ|2 dx, I

M(ψ) =

i 2



¯ x − ψ ψ¯ x ) dx. (ψψ

(1.4)

I

Here I = (0, T0 ). On the other hand, if we consider the more general case of “subharmonic 1 (0, N T ) for some integer N ≥ 2, then the perturbations”, which correspond to q(·, t) ∈ Hper 0 second variation of E at u0 with I = (0, N T0 ) contains additional negative eigenvalues, which cannot be eliminated by restricting the energy to the submanifold where Q and M are constant.

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Generally speaking, in such an unfavorable energy configuration, there is no chance to establish orbital stability using the standard energy method [3]. However, the cubic defocusing NLS equation can (at least formally) be integrated using the inverse scattering transform method, and it admits therefore a countable sequence of independent conserved quantities. For instance, one can verify directly or with an algorithmic computation (see [15, Section 2.3] for a review of such techniques) that the higher-order functional   R(ψ) = I

 1 1 ¯ x + ψ ψ¯ x )2 + |ψ|6 dx, |ψxx |2 + 3|ψ|2 |ψx |2 + (ψψ 2 2

(1.5)

is also invariant under the time evolution defined by (1.1). These additional properties can be invoked to rescue the stability analysis of periodic waves. Indeed, using the eigenfunctions of Lax operators arising in the inverse scattering method, a complete set of Floquet–Bloch eigenfunctions satisfying the linearization of the cubic NLS equation (1.1) at the periodic wave with profile u0 has been constructed in [4]. Moreover, it is shown in [4] that an appropriate linear combination of the energy E, the charge Q, the momentum M, and the higher order quantity R produces a functional for which the periodic wave with profile u0 is a strict local minimizer, up to 2 (0, N T ), for any N ∈ N, where T is the period symmetries. This result holds for q(·, t) ∈ Hper 0 0 of |u0 |. This easily implies that the periodic wave with profile u0 is orbitally stable with respect to subharmonic perturbations. The proof given in [4] that any periodic wave can be characterized as a local minimizer of a suitable higher-order conserved quantity is not direct. Indeed, the authors prove the positivity of the second variation at the periodic wave by evaluating the corresponding quadratic form on the basis of the Floquet–Bloch eigenfunctions associated with the linearized NLS flow. These, however, are not the eigenfunctions of the self-adjoint operator associated with the second variation itself, which would be more natural to use in the present context. In addition, many explicit computations are not transparent because they rely on nontrivial properties of the Jacobi elliptic functions and integrals that are used to represent the profile u0 of the periodic wave. This is why we feel that it is worth revisiting the problem using more standard PDE techniques, which is the goal of the present work. The idea of using higher-order conserved quantities to solve delicate analytical problems related to orbital stability of nonlinear waves in integrable evolution equations has become increasingly popular in recent years. Orbital stability of n-solitons in the Korteweg–de Vries (KdV) and the cubic focusing NLS equations was established in the space H n (R) by combining the first (n + 1) conserved quantities of these equations in [11] and [9], respectively. For the modified KdV equation, orbital stability of breathers in the space H 2 (R) was established in [2] by using two conserved quantities. For the massive Thirring model (a system of nonlinear Dirac equations), orbital stability of solitary waves was proved in the space H 1 (R) with the help of the first four conserved quantities [13]. As already mentioned, we consider in this paper periodic waves of the cubic defocusing NLS equation (1.1) which correspond to real-valued solutions of the second-order equation (1.2). In that case, the second-order equation (1.2) can be integrated once to obtain the first-order equation 

du0 dx

2 =

2

1  1 − u20 − E 2 , 2

x ∈ R,

(1.6)

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Fig. 1. The level set given by (1.6) on the phase plane (u0 , u0 ) for E = 0, 0.4, 0.8.

where the integration constant E ∈ [0, 1] can be used to parameterize all bounded solutions, up to translations. If 0 < E < 1, we obtain a periodic solution which has the explicit form     √ 1+E 1−E u0 (x) = 1 − E sn x , , 2 1+E

(1.7)

where sn(ξ, k) denotes the Jacobi elliptic function with argument ξ and parameter k [10]. This solution corresponds to a closed orbit in the phase plane for (u0 , u0 ), which is represented in Fig. 1. When E → 1 the orbit shrinks to the center point (0, 0), while in the limit E → 0 the solution u0 approaches the black soliton   x u0 (x) = tanh √ , 2

(1.8)

which corresponds to a heteroclinic orbit connecting the two saddle points (−1, 0) and (1, 0). If E ∈ (0, 1), the period of u0 (which is exactly twice the period T0 of the modulus |u0 |) is given by 

2 2T0 = 4 K 1+E



1−E 1+E

 ,

(1.9)

where K(k) is the complete elliptic integral of the first kind. It can be verified that T0 is a decreasing function of E which satisfies T0 → +∞ as E → 0 and T0 → π as E → 1 [7]. Now we study the stability of the periodic wave ψ(x, t) = u0 (x)e−it , where u0 is given by (1.7) for some E ∈ (0, 1). It is clear from (1.2) that the wave profile u0 is a critical point of the energy functional E defined by (1.3). In addition, one can verify by explicit (but rather cumbersome) calculations that u0 is also a critical point of the higher-order functional S(ψ) = R(ψ) −

 1 3 − E 2 Q(ψ), 2

(1.10)

where R is given by (1.5) and Q by (1.4). Using an idea borrowed from [4], we combine E and S by introducing the functional Λc (ψ) = S(ψ) − cE(ψ),

(1.11)

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where c ∈ R is a parameter that will be fixed below. Our first result is the following proposition, which establishes an unconstrained variational characterization for the periodic waves of the NLS equation (1.1), at least when their amplitude is small enough. Proposition 1.1. There exists E0 ∈ (0, 1) such that, for all E ∈ (E0 , 1), there exist values c− and c+ in the range 1 < c− < 2 < c+ < 3 such that, for any c ∈ (c− , c+ ), the second variation of the functional Λc at the periodic wave profile u0 is nonnegative for perturbations in H 2 (R). Furthermore, we have c± = 2 ±



2(1 − E) + O(1 − E) as E → 1.

(1.12)

Remark 1.2. The second variation of Λc at u0 is the quadratic form associated with a fourthorder selfadjoint operator with T0 -periodic coefficients, which will be explicitly calculated in Section 2 below. Proposition 1.1 asserts that the Floquet–Bloch spectrum of that operator is nonnegative, if we consider it as acting on the whole space H 4(R). In particular, the same operator 4 (0, T ), where T is any multiple of T . In fact, has nonnegative spectrum when acting on Hper 0 the proof of Proposition 1.1 shows that Λc (u0 ) is positive except for two neutral directions corresponding to symmetries (translations in space and rotations of the complex phase). This key observation will allow us to prove orbital stability of the periodic wave with respect to subharmonic perturbations, see Theorem 1.8 below. Our second result suggests a rather explicit formula for the limiting values c± that appear in Proposition 1.1. Proposition 1.3. For all E ∈ (0, 1) and all c ≥ 1, the second variation of the functional Λc at the periodic wave profile u0 is positive, except for two neutral directions due to symmetries, only if c ∈ [c− , c+ ] with 2k c± = 2 ± , 1 + k2

 where k =

1−E . 1+E

(1.13)

Remark 1.4. Proposition 1.3 gives a necessary condition for the second variation Λc (u0 ) to be positive except for two neutral directions due to translations and phase rotations. The condition is obtained by considering one particular band of the Floquet–Bloch spectrum of the fourth-order operator associated with Λc (u0 ). That band touches the origin when the Floquet–Bloch wave number is equal to zero, is strictly convex near the origin if c ∈ (c− , c+ ), and is strictly concave if c ≥ 1 and c ∈ / [c− , c+ ]. In the latter case, the second variation Λc (u0 ) has therefore negative directions. Interestingly enough, the alternative approach of Bottman et al. [4] suggests that, for any E ∈ (0, 1), the second variation Λc (u0 ) is positive (except for neutral directions due to symmetries) whenever c ∈ (c− , c+ ). Indeed, after adopting our definition of the functionals E and S, and performing explicit computations with Jacobi elliptic functions, one can show that the conditions implicitly defined in [4, Theorem 7] exactly correspond to choosing our parameter c in the interval (c− , c+ ) given by (1.13). In Fig. 2, the values c± are represented as a function of the parameter E by a solid line. Note that the asymptotic expansion (1.12) is recovered from the analytical expressions (1.13) in the limit k → 0, that is, E → 1. The asymptotic result (1.12) is shown by dashed lines.

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Fig. 2. The values c± given by the explicit expressions (1.13) are represented as a function of the parameter E (solid line). The asymptotic result (1.12) is shown by dashed lines.

The result of Proposition 1.1 relies on perturbation theory and is therefore restricted to periodic waves of small amplitude. Although the analytic formula (1.13) suggests that the conclusion of Proposition 1.1 should hold for all periodic waves, namely for all E ∈ (0, 1), the result of Proposition 1.3 is only a necessary condition for positivity of the functional Λc . In the next result, we fix c = 2 (the mean value in the interval [c− , c+ ]) and prove the positivity of the second variation of the functional Λc=2 . Proposition 1.5. Fix c = 2. For every E ∈ (0, 1), the second variation of the functional Λc=2 at the periodic wave profile u0 is positive, except for two neutral directions due to symmetries. Remark 1.6. In the proof of Proposition 1.5, we show that the quadratic form defined by the second variation Λc=2 (u0 ) restricted to purely imaginary perturbations of the periodic wave can be decomposed as a sum of squared quantities, hence is obviously nonnegative. In order to control the quadratic form for the real perturbations to the periodic wave, we use a continuation argument from the limit to the periodic waves of small amplitude, combined with the analysis of a pair of second-order Schrödinger operators with T0 -periodic coefficients. Remark 1.7. Proposition 1.5 implies the spectral stability of the periodic wave profile u0 for every E ∈ (0, 1), see the end of Section 5. Our final result establishes orbital stability of the periodic wave (1.7) with respect to the sub2 (0, T ), where T > 0 is any integer multiple of the period of u . harmonic perturbations in Hper 0 Therefore, we use I = (0, T ) in the definition of all functionals (1.3)–(1.5). If we consider Λc 2 (0, T ), we know from Proposition 1.5 that Λ (u ) = 0 and that the second as defined on Hper c 0  variation Λc=2 (u0 ) is strictly positive, except for two neutral directions corresponding to symmetries. Since Λc=2 (ψ) is a conserved quantity under the evolution defined by the cubic NLS equation (1.1), we obtain the following orbital stability result. Theorem 1.8. Fix E ∈ (0, 1) and let T be an integer multiple of the period 2T0 of u0 . For any 2 (0, T ) satisfies  > 0, there exists δ > 0 such that, if ψ0 ∈ Hper ψ0 − u0 Hper 2 (0,T ) ≤ δ,

(1.14)

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the unique global solution ψ(·, t) of the cubic NLS equation (1.1) with initial data ψ0 has the following property. For any t ∈ R, there exist ξ(t) ∈ R and θ (t) ∈ R/(2π Z) such that

i(t+θ(t)) 



e ψ · + ξ(t), t − u0 H 2 (0,T ) ≤ . (1.15) per

Moreover ξ and θ are continuously differentiable functions of t which satisfy     ξ˙ (t) + θ˙ (t) ≤ C, t ∈ R,

(1.16)

for some positive constant C. Remark 1.9. It is well known that the Cauchy problem for the cubic NLS equation (1.1) is s (0, T ) for any integer s ≥ 0, see [5]. globally well posed in the Sobolev space Hper Remark 1.10. The proof of Theorem 1.8 shows that, when  ≤ 1, one can take δ = /C for some constant C ≥ 1 depending on E and on the ratio T /T0 . We emphasize, however, that C → ∞ as T /T0 → ∞. This indicates that, although a given periodic wave is orbitally stable with respect to perturbations with arbitrary large period T , the size of the stability basin becomes very small when the ratio T /T0 is large. Applying the same technique, we can also prove the orbital stability of the black soliton (1.8) with respect to perturbations in H 2 (R). The details of this analysis are given in Part II, which is a companion paper to this work. The rest of this article is organized as follows. Section 2 contains the proof of Proposition 1.1. The sufficient condition of Proposition 1.3 is proved in Section 3. In Section 4, we provide a representation of the quadratic form associated with Λc (u0 ) as a sum of squared quantities. Section 5 reports the continuation argument, which yields the proof of Proposition 1.5. Section 6 is devoted to the proof of Theorem 1.8. Appendix A summarizes some explicit computations with the use of Jacobi elliptic functions. 2. Positivity of Λc (u0 ) for periodic waves of small amplitude This section presents the proof of Proposition 1.1. Let u0 be the periodic wave profile defined by (1.7) for some E ∈ (0, 1). We consider perturbations of u0 of the form ψ = u0 + u + iv, where u, v are real-valued. Since u0 is a critical point of both E and S defined by (1.3) and (1.10), the leading order contributions to the renormalized quantities E(ψ) − E(u0 ) and S(ψ) − S(u0 ) are given by the second variations   2  2 2  2  



1   ux + 3u0 − 1 u2 dx + vx + u0 − 1 v 2 dx (2.1) E (u0 )[u, v], [u, v] = 2 I

and  1   S (u0 )[u, v], [u, v] = 2

 I

I

 

u2xx + 5u20 u2x + −5u40 + 15u20 − 4 + 3E 2 u2 dx





+ I



 

2 vxx + 3u20 vx2 + u20 − 1 v 2 dx.

(2.2)

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In the proof of the orbital stability theorem (Theorem 1.8) given in Section 6, we eventually take I = (0, T ), where T is a multiple of the period 2T0 of the periodic wave profile u0 , and 2 (0, T ). In this case, the formulas (2.1) and (2.2) represent the second we assume that u, v ∈ Hper 2 (0, T ). However, here and in the variations of the functionals E and S defined on the space Hper following three sections, we only investigate the positivity properties of the second variations. For that purpose, it is more convenient to take I = R and to assume that u, v ∈ H 2 (R). As is clear from (2.1) and (2.2), the second variations E  (u0 ) and S  (u0 ) are block-diagonal in the sense that the contributions of u and v do not mix together (this is the main reason for which we restrict our analysis to real-valued wave profiles u0 ). We can thus write  1   E (u0 )[u, v], [u, v] = L+ u, u L2 + L− v, v L2 2 and  1   S (u0 )[u, v], [u, v] = M+ u, u L2 + M− v, v L2 , 2 where · ,· L2 is the scalar product on L2 (R) and the operators L± and M± are defined by L+ = −∂x2 + 3u20 − 1,

M+ = ∂x4 − 5∂x u20 ∂x − 5u40 + 15u20 − 4 + 3E 2 ,

L− = −∂x2 + u20 − 1,

M− = ∂x4 − 3∂x u20 ∂x + u20 − 1.

(2.3)

Note that L+ u0 = M+ u0 = 0, due to the translation invariance of the cubic NLS equation (1.1), and that L− u0 = M− u0 = 0, due to the gauge invariance ψ → eiθ ψ with θ ∈ R. We now fix c ∈ R and consider the functional Λc (ψ) = S(ψ) − cE(ψ), as in (1.11). We have      1   Λc (u0 )[u, v], [u, v] = K+ (c)u, u L2 + K− (c)v, v L2 , 2

(2.4)

where K± (c) = M± − cL± . By construction, K± (c) are selfadjoint, fourth-order differential operators on R with T0 -periodic coefficients, where T0 is the period of |u0 |. Our goal is to show that these operators are nonnegative, at least if E is sufficiently close to 1 and if the parameter c is chosen appropriately. Equivalently, the quadratic forms in the right-hand side of (2.4) are nonnegative for all u, v ∈ H 2 (R) under the same assumptions on E and c. Before going further, let us explain why a careful choice of the parameter c is necessary. Assume for simplicity that E = 1, so that u0 = 0. In that case, we have 

 K± (c)u, u L2 =





u2xx − cu2x + (c − 1)u2 dx

R

=

      c 2 c 2 u2 dx. uxx + u dx − 1 − 2 2

R

R

(2.5)

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This simple computation shows that the second variation Λc (0) is nonnegative if and only if c = 2. By a perturbation argument, we shall verify that Λc (u0 ) remains nonnegative for E sufficiently close to 1, provided c is close enough to 2. More precisely, we shall prove that the operators K+ (c) and K− (c) are nonnegative and have only the following zero modes K+ (c)u0 = 0,

and

K− (c)u0 = 0.

(2.6)

This means that the second variation Λc (u0 ) is strictly positive, except along the subspace spanned by the eigenfunctions u0 and iu0 , which correspond to symmetries of the NLS equation (1.1). Note that, when E = 1, the second variation Λc (u0 ) vanishes on a four-dimensional subspace, according to the representation (2.5), but the degeneracy disappears as soon as E < 1. The proof of Proposition 1.1 relies on perturbation theory for the Floquet–Bloch spectrum of the operators K± (c). First, we normalize the period of the profile u0 to 2π by using the transformation u0 (x) = U (x), where  = π/T0 , so that U (z + 2π) = U (z). The second-order differential equation satisfied by rescaled profile U (z), as well as the associated first-order invariant, is given by 2d



2U

dz2

 +U −U =0 3

⇒



2

dU dz

2 =

2

1  1 − U2 − E2 . 2

(2.7)

In agreement with the exact solution (1.7) we assume that U is odd with U  (0) > 0, so that 2 (0, 2π) is entirely determined by the value of E ∈ (0, 1). As was already mentioned, it U ∈ Hper is known for the soft potential in (2.7) that the map (0, 1)  E →  ∈ (0, 1) is strictly increasing and onto [7]. The following proposition specifies the precise asymptotic behavior of the rescaled profile U as E → 1. 2 (0, 2π) can be uniquely described, Proposition 2.1. The map (0, 1)  E → (, U ) ∈ R × Hper when E → 1, by a small parameter a > 0 in the following way:

  E = 1 − a2 + O a4 ,

  3 2 = 1 − a 2 + O a 4 , 4

(2.8)

and  3 U (z) = aU0 (z) + OHper 2 (0,2π) a , where U0 (z) = sin(z). Proof. The argument is rather standard, so we just mention here the main ideas. Since the wave profile U (z) is an odd function of z, we work in the space   L2per,odd (0, 2π) = U ∈ L2loc (R): U is odd and 2π-periodic . ˜ U = aU0 + U˜ , where the perturbation We use the Lyapunov–Schmidt decomposition 2 = 1 + , 2 2 ˜ U ∈ Hper,odd (0, 2π) is orthogonal to U0 in Lper (0, 2π), namely U0 , U˜ L2per = 0. The quantities ˜ and U˜ can be determined by projecting Eq. (2.7) onto the one-dimensional subspace Span{U0 } ⊂ L2per,odd (0, 2π) and its orthogonal complement. This gives the relations

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a ˜ = −

U0 , (aU0 + U˜ )3 L2per

(2.9)

U0 , U0 L2per

and ˜ U˜  + U˜ = (aU0 + U˜ )3 − (1 + )

U0 , (aU0 + U˜ )3 L2per U0 , U0 L2per

U0 .

(2.10)

For any small ˜ and a, it is easy to verify (by inverting the linear operator in the left-hand side 2 and using a fixed point argument) that Eq. (2.10) has a unique solution U˜ ∈ Hper,odd (0, 2π) such 3 ˜ so ˜ ˜ that U0 , U L2 = 0 and U = OH 2 (0,2π) (a ) as a → 0. This solution depends smoothly on , per

per

if we substitute it into the right-hand side of (2.9) we obtain an equation for ˜ only, which can in turn be solved uniquely for small a > 0. The result is ˜ = −a 2

U0 , U03 L2per U0 , U0 L2per

    3 + O a4 = − a2 + O a4 . 4

Finally the expression E = 1 − a 2 + O(a 4 ) follows from the first-order invariant (2.7), if we use the above decompositions and the asymptotic formulas for ˜ and U˜ . 2 We next study the Floquet–Bloch spectrum of the operators K±(c) = M± − cL± . Using the same rescaling z = x and the Floquet parameter κ, we write these operators in the following form   P− (c, κ) = 4 (∂z + iκ)4 − 32 (∂z + iκ)U 2 (∂z + iκ) + c2 (∂z + iκ)2 + (c − 1) 1 − U 2 , P+ (c, κ) = 4 (∂z + iκ)4 − 52 (∂z + iκ)U 2 (∂z + iκ) + c2 (∂z + iκ)2 − 5U 4 + (15 − 3c)U 2 − 4 + 3E 2 + c. Note that the operators P± (c, κ) have π -periodic coefficients, hence we can look for π -periodic Bloch wave functions so that κ can be defined in the Brillouin zone [−1, 1]. However, for computational simplicity of the perturbation expansions, it is more convenient to work with the 2π -periodic Bloch wave functions, in which case κ is defined in the Brillouin zone T = [− 12 , 12 ]. 4 (0, 2π) satisfies If κ ∈ T and if the function w(·, κ) ∈ Hper P± (c, κ)w(z, κ) = λ(κ)w(z, κ),

z ∈ R,

(2.11)

for some λ(κ) ∈ R and either sign, then defining u(x, κ) = eiκx w(x, κ) we obtain a function 4 (R) such that u(·, κ) ∈ L∞ (R) ∩ Hloc K± (c)u(x, κ) = λ(κ)u(x, κ),

x ∈ R.

This precisely means that λ(κ) belongs to the Floquet–Bloch spectrum of K± (c).

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Fig. 3. Left: spectral bands given by (2.12) for c = 2 and a = 0. Right: spectral bands given by the matrix eigenvalue problem (2.16) for c = 2 and a = 0.2.

By Proposition 2.1, when E is close to 1, the operators P± (c, κ) can be expanded as  4 P± (c, κ) = P (0) (c, κ) + a 2 P±(1) (c, κ) + OHper 4 (0,2π)→L2 (0,2π) a , per where P (0) (c, κ) = (∂z + iκ)4 + c(∂z + iκ)2 + c − 1, 3 (1) P− (c, κ) = − (∂z + iκ)4 − 3(∂z + iκ)U02 (∂z + iκ) − 2 3 P+(1) (c, κ) = − (∂z + iκ)4 − 5(∂z + iκ)U02 (∂z + iκ) − 2

3 c(∂z + iκ)2 + (1 − c)U02 , 4 3 c(∂z + iκ)2 + (15 − 3c)U02 − 6. 4

The operator P (0) (c, κ) has constant coefficients, and its spectrum in the space L2per (0, 2π) (0)

consists of a countable family of real eigenvalues {λn (κ)}n∈Z given by 4 2 λ(0) n (κ) = (κ + n) − c(κ + n) + c − 1,

n ∈ Z.

(2.12)

(0)

As was already observed, one has λn (κ) ≥ 0 for all n ∈ Z and all κ ∈ T if and only if c = 2. This (0) is the case represented in Fig. 3 (left), where it is clear that all spectral bands {λn (κ)}κ∈T are strictly positive, except for two bands corresponding to n = ±1 which touch the origin at κ = 0. For small a > 0, the eigenvalues of the perturbed operators P± (c, κ) are denoted by λ± n (κ) (0) ± with n ∈ Z, and we number them in such a way that λn (κ) → λn (κ) as a → 0 for n = ±1. By classical perturbation theory, we know that the eigenvalues λ± n (κ) stay bounded away from ± zero for n = ±1, so it remains to study how the bands {λ± 1 (κ)}κ∈T and {λ−1 (κ)}κ∈T behave near κ = 0 as a → 0. The following proposition indicates that these bands separate from each other when a > 0, so that one band still touches the origin at κ = 0 while the other one remains strictly positive for all κ ∈ T. In other words, the degeneracy of the limiting case c = 2, a = 0 is unfold by the perturbation as soon as a > 0. This phenomenon is illustrated in Fig. 3 (right), which shows the solutions of the matrix eigenvalue problem (2.16) obtained below. Proposition 2.2. If a > 0 is sufficiently small and c ∈ (c− , c+ ), where c± = 2 ±

√   2a + O a 2 ,

(2.13)

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the operator K± (c) has exactly one Floquet–Bloch band denoted by {λ± touches −1 (κ)}κ∈T that √ the origin at κ = 0, while all other bands are strictly positive. Moreover, for any ν < 2, there exist positive constants C1 , C2 , C3 (independent of a) such that, if |c − 2| ≤ νa, one has 2 λ± −1 (κ) ≥ C1 κ ,

 2  2 λ± 1 (κ) ≥ C2 a + κ ,

λ± n (κ) ≥ C3 ,

n ∈ Z \ {+1, −1},

(2.14)

for all κ ∈ T. Proof. From (2.12) we know that, if |c − 2| is sufficiently small, there exists a constant C > 0 (independent of c) such that −1 0 < λ(0) ≤C n (κ)

for all n ∈ Z \ {+1, −1} and all κ ∈ T.

(2.15)

By classical perturbation theory, this bound remains true (with possibly a larger constant C) for the perturbed eigenvalues λ± n (κ) when n = ±1 and a > 0 is small enough. We thus obtain the third estimate in (2.14). To control the critical bands corresponding to n = ±1, we concentrate on the operator P− (c, κ) (the argument for P+ (c, κ) being similar, see below), and for simplicity we denote its eigenvalues by λn (κ) instead of λ− n (κ). The same perturbation argument as before shows that λ±1 (κ) is bounded away from zero if |κ| ≥ κ0 and a is sufficiently small, where κ0 > 0 is an arbitrary positive number. On the other hand, for small values of a, |c − 2|, and |κ|, solutions to the spectral problem (2.11) for P− (c, κ) are obtained by the Lyapunov–Schmidt decomposition w(z, κ) = b1 (κ)eiz + b−1 (κ)e−iz + w(z, ˜ κ),



 e±i· , w(·, ˜ κ) L2 = 0, per

where all terms can be determined by projecting the spectral problem (2.11) onto the twodimensional subspace Span{ei· , e−i· } ⊂ L2per (0, 2π) and its orthogonal complement in 2 ˜ κ) = OHper L2per (0, 2π). Using the bound (2.15), one can prove that w(·, 4 (0,2π) (a ), which al(0)

lows us to find λ(κ) near λ±1 (κ) as a solution of the matrix eigenvalue problem 

(0)

λ1 (κ) + a 2 g1,1 (κ) + O(a 4 ) −a 2 g−1,1 (κ) + O(a 4 )   b1 = λ(κ) , b−1

−a 2 g1,−1 (κ) + O(a 4 ) 2 4 λ(0) −1 (κ) + a g−1,−1 (κ) + O(a )



b1 b−1

where 3 3 1 3 g±1,±1 (κ) = − (κ ± 1)4 + c(κ ± 1)2 + (1 − c) + (κ ± 1)2 , 2 4 2 2  1 3 2 g±1,∓1 (κ) = (1 − c) + κ − 1 . 4 4 Setting c = 2 + γ with small |γ |, we have



(2.16)

T. Gallay, D. Pelinovsky / J. Differential Equations 258 (2015) 3607–3638

3619

(0)

λ±1 (κ) = ∓2γ κ + (4 − γ )κ 2 ± 4κ 3 + κ 4 , 1 3 3 3 g±1,±1 (κ) = 1 + γ ± γ κ − 6κ 2 + γ κ 2 ∓ 6κ 3 − κ 4 , 4 2 4 2 1 3 g±1,∓1 (κ) = −1 − γ + κ 2 . 4 4 If we denote by A the matrix in the left-hand side of (2.16), we thus obtain the expansions    1 tr(A) = a 2 + 4κ 2 + O a 2 + κ 2 |γ | + a 2 + κ 2 , 2  2 2    det(A) = a 2 + 4κ 2 − a 4 − 4γ 2 κ 2 + O a 2 + κ 2 |γ | + a 2 + κ 2 . As a result, the eigenvalues λ±1 (κ) of A satisfy    λ±1 (κ) = a 2 + 4κ 2 + O a 2 + κ 2 |γ | + a 2 + κ 2  2    ± a 4 + 4γ 2 κ 2 + O a 2 + κ 2 |γ | + a 2 + κ 2 .

(2.17)

It remains to analyze (2.17). If a > 0 is small, we obviously have    λ1 (κ) ≥ a 2 + 4κ 2 + O a 2 + κ 2 |γ | + a 2 + κ 2 > 0, which implies the second bound in (2.14). To estimate λ−1 (κ), we first consider the regime where |κ| ≤ a. If |γ | ≤ νa for any ν > 0 independently of a, further expansion of (2.17) yields λ−1 (κ) = μ + 4κ 2 −

   2γ 2 κ 2 + O κ 2 |γ | + a 2 , 2 a

(2.18)

where μ = O(a 2 (|γ | + a 2 )) does not depend on κ. But since K− (c)u0 = 0 for any c, we must have λ−1 (0) = 0 to all orders in a and γ , hence actually μ = 0. Then (2.18) shows that λ−1 (κ) has a nondegenerate minimum at κ = 0 if and only if   γ 2 < 2a 2 + O a 3 .

(2.19)

Since γ = c − 2,√this yields expansion (2.13) for c± . From now on, we assume that |γ | ≤ νa for some ν ∈ (0, 2), so that the inequality (2.19) certainly holds if a is sufficiently small. The expansion (2.18) shows that if |κ| ≤ a, then      λ−1 (κ) = 4 − 2ν 2 κ 2 + O κ 2 |γ | + a 2 . On the other hand, if |κ| ≥ a, we easily find from (2.17) that 1/2  1/2      ≥ κ 2 + O κ 2 |γ | + κ 2 , λ−1 (κ) ≥ 4κ 2 − 2|γ ||κ| + O κ 2 |γ | + κ 2 because 2|γ ||κ| ≤ κ 2 + γ 2 ≤ κ 2 + ν 2 a 2 ≤ 3κ 2 . Altogether, we obtain the first estimate in (2.14).

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The spectral problem (2.11) for the operator P+ (c, κ) can be studied in a similar way and results in the matrix eigenvalue problem (2.16) with 3 3 3 5 g±1,±1 (κ) = − (κ ± 1)4 + c(κ ± 1)2 + (1 − c) + (κ ± 1)2 , 2 4 2 2  3 5 g±1,∓1 (κ) = (5 − c) + κ 2 − 1 . 4 4 Although the matrix A has now different entries, the leading order terms for the quantities tr(A) and det(A) are unchanged, hence the eigenvalues λ±1 (κ) still satisfy (2.17). Consequently, the conclusion remains true for c in the same interval (2.13). 2 Remark 2.3. In view of expansion (2.8), Proposition 1.1 is a direct consequence of Proposition 2.2. 3. Necessary condition for positivity of Λc (u0 ) This section presents the proof of Proposition 1.3. In Section 2, we only considered small amplitude periodic waves (1.7) with E close to 1. To get some information on the quadratic form Λc (u0 ) for larger periodic waves, we recall that, for any E ∈ (0, 1) and any c ∈ R, the operators P± (c, κ) have at least one Floquet–Bloch spectral band that touches the origin at κ = 0, because we know from (2.6) that the kernel of P± (c, 0) in L2per (0, 2π) is nontrivial. In what follows, we focus on the operator P− (c, κ). Assuming that ker(P− (c, 0)) in L2per (0, 2π) is one-dimensional, we compute an asymptotic expansion as κ → 0 of the unique Floquet–Bloch band that touches the origin at κ = 0. By Proposition 2.2, the assumption on ker(P− (c, 0)) is satisfied at least for the periodic waves of small amplitude, in which case the Floquet–Bloch band that touches the origin is actually the lowest band λ− −1(κ). Proposition 3.1. Fix E ∈ (0, 1) and assume that U = u0 (−1 ·) is the only 2π -periodic solution of the homogeneous equation P− (c, 0)w = 0 for some c ∈ R. Denote by μ(κ) the Floquet–Bloch band of P− (c, κ) that touches zero at κ = 0. Then μ is C 2 near κ = 0, μ(0) = μ (0) = 0, and μ (0) =

  −1  2 −44 (c − 2)2 U  , P− (c, 0) U  L2 2 per U L2 per

2

+ 34 U  L2 + (3 − c)2 U 2L2 , per

per

(3.1)

where W =(P− (c, 0))−1 U  is uniquely defined under the orthogonality condition U, W L2per =0. 4 (0, 2π). Proof. We consider P− (c, κ) as a self-adjoint operator in L2per (0, 2π) with domain Hper As κ → 0, we have

 3 , P− (c, κ) = P0 (c) + iκP1 (c) − κ 2 P2 (c) + OHper 4 (0,2π)→L2 (0,2π) κ per

T. Gallay, D. Pelinovsky / J. Differential Equations 258 (2015) 3607–3638

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where   P0 (c) = 4 ∂z4 − 32 ∂z U 2 ∂z + c2 ∂z2 + (c − 1) 1 − U 2 , P1 (c) = 44 ∂z3 − 62 U 2 ∂z − 62 U U  + 2c2 ∂z , P2 (c) = 64 ∂z4 − 32 U 2 + c2 . We note that P0 (c) and P2 (c) are self-adjoint, whereas P1 (c) is skew-adjoint. Under the assumptions of the proposition, the Floquet–Bloch band μ(κ) that touches zero at κ = 0 is separated from all the other bands of P− (c, κ) locally near κ = 0. Thus, μ(κ) is smooth near κ = 0, and it is possible to choose a nontrivial solution w(z, κ) of the eigenvalue equation P− (c, κ)w(z, κ) = μ(κ)w(z, κ) which also depends smoothly on κ. We look for an expansion of the form   μ(κ) = iκμ1 − κ 2 μ2 + O κ 3 and  3 , w(z, κ) = U (z) + iκw1 (z) − κ 2 w2 (z) + OHper 4 (0,2π) κ where w1 , w2 , and the remainder term belong to the orthogonal complement of span{U } in L2per (0, 2π). This gives the following system for the correction terms P0 (c)w1 + P1 (c)U = μ1 U,

(3.2)

P0 (c)w2 + P1 (c)w1 + P2 (c)U = μ1 w1 + μ2 U.

(3.3)

If we take the scalar product of (3.2) with U in L2per (0, 2π) and use the fact that P0 (c) is selfadjoint, P1 (c) is skew-adjoint, and P0 (c)U = 0, we obtain μ1 = 0. Similarly, taking the scalar product of (3.3) with U gives a nontrivial equation for μ2 :     μ2 U 2L2 = U, P1 (c)w1 L2 + U, P2 (c)U L2 . per

per

per

We note that   P1 (c)U = 22 22 U  − 6U 2 U  + cU  = 22 (c − 2)U  ,     P2 (c)U = 2 62 U  − 3U 3 + cU = 2 32 U  + (c − 3)U . Setting w1 = −22 (c − 2)W , where W is the unique solution of P0 (c)W = U  subject to the orthogonality condition U, W L2per = 0, we obtain     2 μ2 U 2L2 = 44 (c − 2)2 U  , W L2 − 34 U  L2 + (3 − c)2 U 2L2 , per

per

which yields the result (3.1) since μ (0) = −2μ2 .

per

2

per

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Note that the first term in the right-hand side of (3.1) is negative, whereas the other two are positive for c ≤ 3. In the particular case where c = 2, it follows from Lemma 4.1 below that ker(P− (2, 0)) = span{U } for any value of the parameter E ∈ (0, 1), so that the assumption of Proposition 3.1 is satisfied. In this case, the formula (3.1) shows that μ (0) > 0. Next, we give an explicit expression for μ (0) by evaluating the various terms in (3.1) using known properties of the Jacobi elliptic functions. These computations are performed in Appendix A, see Eqs. (A.8)–(A.12), and yield the explicit formula μ (0) =

22 k 2 (4k 2 − (c − 2)2 (1 + k 2 )2 ) (1 + k 2 )(1 −

E(k) 2 K(k) )(2k

+ (c − 2)(1 + k 2 )(1 −

E(k) K(k) ))

,

(3.4)

where K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively, and the parameter k ∈ (0, 1) is given by (1.13). The denominator in (3.4) is strictly positive if c ≥ 1. Indeed, since K(k) > E(k) for all k ∈ (0, 1), thanks to Eq. (A.10) in Appendix A, the denominator in (3.4) is a strictly increasing function of c, and for c = 1 we have       E(k) E(k)  2k 2 + (c − 2) 1 + k 2 1 − = k2 − 1 + k2 + 1 > 0.  K(k) c=1 K(k) The expression above is positive for all k ∈ (0, 1), thanks to Eq. (A.11) in Appendix A. Thus, for c ≥ 1, the sign of μ (0) is the sign of the numerator in (3.4). It follows that μ (0) ≥ 0 if c ∈ [c− , c+ ] ⊂ [1, 3], where c± are given by (1.13). Similarly, we have μ (0) < 0 if c ≥ 1 and c∈ / [c− , c+ ]. Remark 3.2. The computations above imply the conclusion of Proposition 1.3. Indeed, either the kernel of P− (c, 0) in L2per (0, 2π) is one-dimensional, in which case the perturbation argument of Proposition 3.1 applies and proves the existence of negative spectrum if c ≥ 1 is outside [c− , c+ ], or the kernel is higher-dimensional and the second variation Λc (u0 ) has more neutral directions than the two directions due to the symmetries. Note that we do not claim that the second variation Λc (u0 ) (or even the quadratic form associated with K− (c)) is indeed positive if c ∈ (c− , c+ ), although by Proposition 2.2 this is definitely the case for the periodic waves of small amplitudes. Remark 3.3. If we compare the above results with the computations in [4], one advantage of our approach is that we clearly distinguish between the spectra of the two linear operators K+(c) and K− (c). In particular, the necessary condition in Proposition 1.3 is derived from the positivity of the Floquet–Bloch spectrum of K− (c). We expect that, for any E ∈ (0, 1), the Floquet–Bloch spectrum of K+ (c) is positive for c in a larger subset of R than (c− , c+ ). For instance, the operator K+ (c) is positive in L2 (R) for every c ≤ 3 in the case of the black soliton that corresponds to E = 0, see Remark 4.6 below. 4. Positive representations of Λc (u0 ) As a first step in the proof of Proposition 1.5, which claims that the quadratic forms associated with the linear operators K± (c) are nonnegative on H 2 (R) if c = 2, we look for representations of these quadratic forms as sums of squared quantities. Our first result shows that, if c = 2, the quadratic form associated with K− (c) is always positive, for all E ∈ [0, 1], including the black soliton for E = 0 and the zero solution for E = 1.

T. Gallay, D. Pelinovsky / J. Differential Equations 258 (2015) 3607–3638

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Lemma 4.1. Fix c = 2. For any E ∈ [0, 1] and any v ∈ H 2 (R), we have 

K− (2)v, v

 L2

2 = L− v 2L2 + u0 vx − u0 v L2 .

(4.1)

Proof. Using the definition (2.3) of the operator L− and integrating by parts, we obtain  2  2

  2 vxx + 2 1 − u20 vvxx + 1 − u20 v 2 dx L− v L2 = R



=



  2

   2 vxx − 2 1 − u20 vx2 − 2 u0 u0 v 2 + 1 − u20 v 2 dx.

R

Similarly, we obtain

u0 vx − u v 2 2 = 0 L





   2

u20 vx2 + u0 u0 v 2 + u0 v 2 dx.

R

As a consequence, we have

2 L− v 2L2 + u0 vx − u0 v L2 =





2



   2 vxx + 3u20 − 2 vx2 + 1 − u20 − u0 u0 v 2 dx,

R

which yields the desired result since (1 − u20 )2 − u0 u0 = 1 − u20 .

2

Remark 4.2. It is easy to verify that the right-hand side of the representation (4.1) vanishes if and / H 2 (R), this shows that K− (2)v, v L2 > 0 for any only if v = Cu0 for some constant C. As u0 ∈ 2 nonzero v ∈ H (R). Unfortunately, we are not able to find a positive representation for the quadratic form associated with the operator K+ (c). If we proceed as in the proof of Lemma 4.1, we obtain   2 2

 K+ (2)u, u L2 = L+ u 2L2 − u0 ux − 3u20 u2 + 5u40 u2 dx. (4.2) R

Here the second term in the right-hand side has no definite sign, hence it is difficult to exploit the representation (4.2). In the following lemma, we give a partial result which shows that the quadratic form associated with K+ (c) is positive for c < 3 at least on a subspace of H 2 (R). Lemma 4.3. For any E ∈ (0, 1), any c ∈ R, and any u ∈ H 2 (R) such that u(x) = 0 whenever u0 (x) = 0, we have

u0 w 2 

K+ (c)u, u L2 = wx 2L2 + (3 − c) w 2L2 + 2E 2

u 2 , 0 L



where w = ux −

u0 u ∈ H 1 (R) u0

satisfies

w u0

∈ L2 (R).

(4.3)

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Proof. Since u0 satisfies the second-order differential equation L+ u0 = 0, the zeros of u0 are all simple, as can also be deduced from the explicit formula (1.7). Thus, if u ∈ H 2 (R) is such that u(x) = 0 whenever u0 (x) = 0, we can write u = u0 u˜ and it follows from Hardy’s inequality that u˜ ∈ H 1 (R). With this notation, we have w := ux −

u0 u = ux − u0 u˜ = u0 u˜ x , u0

so that w ∈ H 1 (R) and uw ∈ L2 (R). As a consequence, all terms in right-hand side of (4.3) are 0 well-defined, and the integrations by parts used in the computations below can easily be justified. To prove the representation (4.3), we first note that   u u uxx + 1 − 3u20 u = uxx − 0 u = wx + 0 w. u0 u0 Integrating by parts, we thus obtain L+ u 2L2

  

 2 2(u0 )2 2  u0 2 2 2

= wx +  w = wx L2 + w dx. 1 − 3u0 w + u0 L2 (u0 )2 R

On the other hand, we have  w 2L2



 

u2x + 3u20 − 1 u2 dx,

= R

and  u0 w 2L2 =



  u20 u2x + 5u20 − 3 u20 u2 dx.

R

Thus, using the analogue of (4.2) for all c ∈ R, we find 

 K+ (c)u, u L2 = L+ u 2L2 + (2 − c)





u2x − u2 + 3u20 u2 dx −

R

 

= wx 2L2 + (3 − c) w 2L2 + 2 R

which yields the desired result since

(u0 )2 (u0 )2





u20 u2x − 3u20 u2 + 5u40 u2 dx

R

(u0 )2 (u0 )2

 2 − 2u0 w 2 dx,

u2

− 2u20 = E 2 (u0)2 holds by Eqs. (1.2) and (1.6). 0

2

Remark 4.4. If c ≤ 3, the right-hand side of the representation (4.3) is nonnegative and vanishes if and only if w = 0, which is equivalent to u = Cu0 for some constant C. However, this does not imply positivity of the quadratic form associated with K+ (c), because the representation (4.3) only holds for u in a subspace of H 2 (R). As a matter of fact, the right-hand side of the

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3625

representation (4.3) is positive for any c ≤ 3, whereas we know from the proof of Proposition 2.2 that, when E is close to 1, the operator K+ (c) is positive if and only if c ∈ (c− , c+ ) where c± → 2 as E → 1. For the black soliton (1.8) corresponding to the case E = 0, the proof of Lemma 4.3 yields a much stronger conclusion, because u0 never vanishes so that we do not need to impose any √ restriction to u ∈ H 2 (R). Using the identity u0 = − 2u0 u0 which holds for the black soliton (1.8) only, we obtain the following result. Corollary 4.5. Consider the black soliton (1.8), for which E = 0. For any c ∈ R and any u ∈ H 2 (R), we have 

 K+ (c)u, u L2 = wx 2L2 + (3 − c) w 2L2 ,

where w = ux +

(4.4)

√ 2u0 u ∈ H 1 (R).

Remark 4.6. If c ≤ 3, the right-hand side of the representation (4.4) is nonnegative and vanishes if and only if w = 0, which is equivalent to u = Cu0 for some constant C. Note that u0 ∈ H 2 (R) in the present case. On the other hand, using definitions (2.3) and the fact that u0 (x) → ±1 as x → ±∞, it is easy to verify that K+ (c) has some negative essential spectrum as soon as c > 3. Thus the representation (4.4) gives a sharp positivity criterion for the operator K+ (c) in the case of the black soliton (1.8). 5. Positivity of Λc=2 (u0 ) for periodic waves of large amplitude This section presents the proof of Proposition 1.5. The energy functionals (1.3) and (1.10) generate two different flows in the hierarchy of integrable NLS equations, see [4]. If we consider E and S as functions of the complex variables ψ ¯ these flows are defined by the evolution equations and ψ,

i

∂ψ δE , = ∂t δ ψ¯

i

∂ψ δS , = ∂τ δ ψ¯

(5.1)

where the symbol δ is used to denote the standard variational derivative. Here t is the time of the cubic defocusing NLS equation (1.1), whereas τ is the time of the higher-order NLS equation. Since the quantities E and S are in involution, the flows defined by both equations in (5.1) commute with each other. In what follows, we fix some E ∈ (0, 1) and consider the periodic wave profile u0 defined by (1.7). Using the real-valued variables u, v for the perturbations, as in the representations (2.1) and (2.2), we obtain the following evolution equations for the linearized flows of the cubic NLS equation and the higher-order NLS equation at the periodic wave profile u0 : ∂ ∂t

  u v

 =

0 −L+

L− 0

  u v

,

∂ ∂τ

  u v

 =

0 −M+

M− 0

  u v

,

(5.2)

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where the operators L± and M± are given by (2.3). Because the linearized flows also commute with each other, the operators L± and M± satisfy the following intertwining relations L− M+ = M − L+ ,

L+ M− = M + L− .

(5.3)

Of course, the relations (5.3) can also be verified by a direct calculation, using the differential equations (1.2) and (1.6) satisfied by the periodic wave profile u0 . It follows from the relation (5.3) that, for every c ∈ R, we have L− K+ (c) = K− (c)L+ ,

L+ K− (c) = K+ (c)L− ,

(5.4)

where K± (c) = M± − cL± as before. Given the positivity of the operator K− (2) established in Lemma 4.1, we shall use the intertwining relations (5.4) to deduce the positivity of the operator K+ (2). This is achieved by studying all bounded solutions of the homogeneous equations associated with operators L± and K± (2) and by applying a continuation argument from the limit E → 1, where positivity of the operator K+ (2) is proved in Proposition 2.2. 2 (R) satisfies L u = 0, then u = Cu for some constant C. Lemma 5.1. If u ∈ L∞ (R) ∩ Hloc + 0 2 Moreover, there exists a unique odd, 2T0 -periodic function U ∈ Hper,odd (0, 2T0 ) such that L+ U = u0 , where 2T0 is the period of u0 .

Proof. We know that L+ u0 = 0. Another linearly independent solution to the equation L+v = 0 can be obtained by differentiating the periodic wave profile u0 with respect to the parameter E ∈ (0, 1), namely v = ∂E u0 . Indeed, if we differentiate the equation u0 + u0 − u30 = 0 with respect to the parameter E , we see that   L+ v = −v  + 3u20 − 1 v = 0. Moreover, v(x) is an odd function of x that grows linearly as |x| → ∞. The latter claim can be verified by differentiating the explicit formula (1.7) with respect to E , but that calculation is not immediate because it involves the derivative of the Jacobi elliptic function sn(ξ, k) with respect to the parameter k. Alternatively, we can use Floquet theory to deduce that v is either periodic of period 2T0 , where 2T0 is the minimal period of u0 , or grows linearly at infinity. The first possibility is excluded by the following argument. If we denote u0 (x) = u0 (x; E) and T0 = T0 (E) to emphasize the dependence upon the parameter E , we have by construction   u0 (0; E) = u0 2T0 (E); E = 0. Differentiating that relation with respect to E, we find v(0) = 0

and v(2T0 ) + 2u0 (2T0 )T0 (E) = 0.

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But we know that u0 (2T0 ) = u0 (0) > 0 and that T0 (E) < 0, hence we deduce that v(2T0 ) > 0, which implies that v is not periodic of period 2T0 . This proves that the kernel of L+ (in the space of bounded functions) is spanned by u0 , which is the first part of the statement. For the second part of the statement, we look for solutions of the inhomogeneous equation L+ U = u0 and note that the Fredholm solvability condition u0 , u0 L2per = 0 is trivially satisfied in the space of 2T0 -periodic functions. Hence, there exists a unique odd 2T0 -periodic solution U of the inhomogeneous equation L+ U = u0 in the domain of L+ , that is, U ∈ 2 Hper,odd (0, 2T0 ). 2 2 (R) satisfies L v = 0, then v = Cu for some constant C. Lemma 5.2. If v ∈ L∞ (R) ∩ Hloc − 0 2 (0, 2T0 ) such that Moreover, there exists a unique even, 2T0 -periodic function V ∈ Hper,even L− V = u0 , where 2T0 is the period of u0 .

Proof. We know that L− u0 = 0. Another linearly independent solution to the equation L−v = 0 is given by v(x) = 2u0 (x) − u0 (x)

x u0 (y)2 dy,

x ∈ R,

0

as is easily verified by a direct calculation. Clearly v(x) is an even function of x that grows linearly as |x| → ∞. This proves that the kernel of L− (in the space of bounded functions) is spanned by u0 . The second part of the statement follows by the same argument as in the proof of Lemma 5.1. 2 Remark 5.3. The solutions U and V of the inhomogeneous equations L+ U = u0 and L− V = u0 can be expressed explicitly in terms of the Jacobi elliptic functions, see Eqs. (A.5) and (A.17) in Appendix A. Next, we establish analogues of Lemmas 5.1 and 5.2 for the operators K± (c) in the particular case c = 2. 4 (R) satisfies K (2)v = 0, then v = Cu for some constant C. Lemma 5.4. If v ∈ L∞ (R) ∩ Hloc − 0

Proof. Using integration by parts as in the proof of Lemma 4.1, we obtain the following identity for any v ∈ H 4 (−N T0 , N T0 ), where N ∈ N and 2T0 is the period of u0 : N T0 

N T0 

vK− (2)v dx = −N T0

 2   

x=N T |L− v|2 + u0 vx − u0 v  dx − 2 1 − u20 vvx + u0 u0 v 2 x=−N0T .



0

−N T0

4 (R) satisfies K (2)v = 0. By standard elliptic estimates, we Assume now that v ∈ L∞ (R) ∩ Hloc − know that v is smooth on R and that all derivatives of v are bounded. Moreover, since the operator K− (2) has T0 -periodic coefficients, it follows from Floquet theory that v(x) = eiγ x w(x), where γ ∈ R and w is smooth on R and T0 -periodic. Using the identity above, we thus obtain

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1 0= N

N T0  −N T0

T0 = −T0



 2  

x=N T 1  |L− v|2 + u0 vx − u0 v  dx − 2 1 − u20 vvx + u0 u0 v 2 x=−N0T 0 N



 2  

x=N T 1  |L− v|2 + u0 vx − u0 v  dx − 2 1 − u20 vvx + u0 u0 v 2 x=−N0T . 0 N

Taking the limit N → ∞ and using the boundedness of v and vx , we obtain L− v = 0 and u0 vx − u0 v = 0 for all x ∈ R. By Lemma 5.2, we conclude that v = Cu0 for some constant C. 2 4 (R) satisfies K (2)u = 0, then u = Cu for some constant C. Lemma 5.5. If u ∈ L∞ (R) ∩ Hloc + 0 4 (R) satisfies K (2)u = 0. By the intertwining relaProof. Assume that u ∈ L∞ (R) ∩ Hloc + tion (5.4), we have K− (2)L+ u = L− K+ (2)u = 0. Using Lemma 5.4, we deduce that L+ u = Bu0 for some constant B. Finally, Lemma 5.1 implies that u = BU + Cu0 for some constant C. In particular, we have 0 = K+ (2)u = BK+ (2)U , because K+ (2)u0 = 0. Now an explicit computation that is carried out in Appendix A shows that K+ (2)U = Du0 for some constant D = 0, see Eq. (A.19), so that K+ (2)U is not identically zero. Thus B = 0, hence u = Cu0 . 2

Remark 5.6. The result of Lemma 5.5 yields the conclusion of Proposition 1.5. Indeed, in the limit E → 1, positivity of the operator K+ (2) is proved in Proposition 2.2. All Floquet–Bloch bands are strictly positive, except for the lowest band that touches the origin because of the zero eigenvalue due to translational symmetry, see Fig. 3. When the parameter E is decreased from 1 to 0, the Floquet–Bloch spectrum of K+ (2) evolves continuously, and positivity of the spectrum is therefore preserved as long as no other band touches the origin. Such an event would result in the appearance of another bounded solution to the homogeneous equation K+(2)u = 0, besides the zero mode u0 due to translation invariance. By Lemma 5.5, such a solution does not exist, hence K+ (2) is a nonnegative operator for any E ∈ (0, 1). To conclude this section, we note that the intertwining relations (5.4) and the positivity of the operators K± (2) established in Proposition 1.5 imply the spectral stability of the periodic wave. Consider the linearized operator with T0 -periodic coefficients given by  J L :=

0

1

−1 0



L+

0

0

L−



 =

0 −L+

 L− , 0

(5.5)

and acting on vectors in L2 (R) × L2 (R). We say that the periodic wave is spectrally stable if the Floquet–Bloch spectrum of J L is purely imaginary. Let λ ∈ C belong to the Floquet–Bloch spec˜ trum, so that J Lψ = λψ for some nonzero eigenfunction ψ . We know that ψ(x) = eiγ x ψ(x), ˜ where γ ∈ R and ψ is T0 -periodic. We want to show that λ ∈ iR. Let K := diag[K+ (2), K− (2)]. Then J LJ Kψ = J KJ Lψ = λJ Kψ, because the operators J L and J K commute due to the intertwining relations (5.4). As J is invertible, we thus have LJ Kψ = λKψ . If we now take the scalar product of both sides with the eigenfunction ψ in the space L2 (0, T0 ) × L2 (0, T0 ), we obtain ¯ λψ, Kψ L2 = ψ, LJ Kψ L2 = −J Lψ, Kψ L2 = −λψ, Kψ L2 ,

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where we have used the fact that L is self-adjoint and J is skew-adjoint. If λ = 0, then ψ is not a linear combination of the two neutral eigenfunctions (u0 , 0) and (0, u0 ). In that case, we have ¯ that is, λ ∈ iR. ψ, Kψ L2 > 0 by Proposition 1.5, and the identity above shows that λ = −λ, Remark 5.7. Spectral stability of the periodic wave is established in [4], where explicit expressions for the Floquet–Bloch spectrum of the operator J L and the associated eigenfunctions are obtained using Jacobi elliptic functions. In our approach, once positivity of the operator K is known, the spectral stability of the periodic wave follows from the commutativity of the operators J L and J K and is established by a general argument that does not use the specific form of the eigenfunctions. 6. Proof of orbital stability of a periodic wave This section is devoted to the proof of Theorem 1.8. We fix E ∈ (0, 1) and consider the periodic wave profile u0 given by (1.7). Let T be a multiple 2 (0, T ) is close of the period 2T0 of u0 , so that T = 2N T0 for some integer N ≥ 1. If ψ0 ∈ Hper 2 (0, T )) of to u0 in the sense of the initial bound (1.14), we claim that the solution ψ ∈ C(R, Hper the cubic NLS equation (1.1) with initial data ψ0 can be characterized as follows. For any t ∈ R, there exist modulation parameters ξ(t) ∈ R and θ (t) ∈ R/(2π Z) such that   eit+iθ(t) ψ x + ξ(t), t = u0 (x) + u(x, t) + iv(x, t),

x ∈ R,

(6.1)

2 (0, T ) are real-valued functions satisfying the orthogonality condiwhere u(·, t), v(·, t) ∈ Hper tions



 u0 , u(·, t) L2 = 0, per



 u0 , v(·, t) L2 = 0, per

(6.2)

where · ,· L2per denotes the usual scalar product in L2per (0, T ). Note that the orthogonality conditions (6.2) are not symplectic orthogonality conditions for the NLS equation, in contrast with the conditions that are often used to study the asymptotic stability of nonlinear waves [12]. To prove the decomposition (6.1), we proceed in two steps. We first show that the representation (6.1) holds whenever ψ(·, t) is sufficiently close to the orbit of u0 under translations and phase rotations. 2 (0, T ) Lemma 6.1. There exists constants 0 ∈ (0, 1) and C0 ≥ 1 such that, for any ψ ∈ Hper satisfying

d := inf eiθ ψ(· + ξ ) − u0 H 2 ≤ 0 , ξ,θ∈R

per

(6.3)

one can find modulation parameters ξ ∈ R and θ ∈ R/(2π Z) such that eiθ ψ(x + ξ ) = u0 (x) + u(x) + iv(x),

x ∈ R,

(6.4)

2 (0, T ) satisfy the orthogonality conditions (6.2) and d ≤ u + iv where u, v ∈ Hper 2 ≤ C0 d. Hper

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Proof. We consider the smooth function f : R2 → R2 defined by  f(ξ, θ ) =

u0 (· − ξ ), Re(eiθ ψ) L2per



u0 (· − ξ ), Im(eiθ ψ) L2per

.

We have f(ξ, θ ) = 0 if and only if ψ can be represented as in the decomposition (6.4) with u, v satisfying the orthogonality conditions (6.2). Let (ξ0 , θ0 ) ∈ R2 denote the arguments of the infimum in (6.3) (note that one can restrict the values of (ξ, θ ) to [0, T ] × [0, 2π ], so that the minimum exists). Then assumption (6.3) implies that f(ξ0 , θ0 ) ≤ Cd, for some constant C independent of ψ . On the other hand, the Jacobian matrix of the function f at the point (ξ0 , θ0 ) is given by Df(ξ0 , θ0 ) =

 u 2

0 L2per

0  +



0 u0 2L2

per

−u0 , Re(eiθ0 ψ(· + ξ0 ) − u0 ) L2per

−u0 , Im(eiθ0 ψ(· + ξ0 ) − u0 ) L2per

−u0 , Im(eiθ0 ψ(· + ξ0 ) − u0 ) L2per

u0 , Re(eiθ0 ψ(· + ξ0 ) − u0 ) L2per

 .

The first term in the right-hand side is a fixed invertible matrix and the second term is bounded in norm by Cd, hence Df(ξ0 , θ0 ) is invertible if 0 is small enough, with (Df(ξ0 , θ0 ))−1 ≤ C where C is a positive constant independent of ψ . Finally, it is straightforward to verify that the second order derivative of f is uniformly bounded if 0 < 1. These observations together imply that there exists a unique pair (ξ, θ ), in the O(d) neighborhood of the point (ξ0 , θ0 ), such that f(ξ, θ ) = 0. Thus, we have the decomposition (6.4) with these values of (ξ, θ ), and

iθ

−iθ u + iv Hper u0 (· − ξ ) H 2 2 = e ψ(· + ξ ) − u0 2 = ψ − e Hper per



≤ ψ − e−iθ0 u0 (· − ξ0 ) H 2 + e−iθ0 u0 (· − ξ0 ) − e−iθ u0 (· − ξ ) H 2 per

per

≤ C0 d, where C0 ≥ 1 is independent of ψ . This concludes the proof. 2 We next show that the solution ψ(·, t) of the cubic NLS equation (1.1) stays close to the orbit of u0 for all times. To show this, we use the conserved quantity Λc given by (1.11), where it is understood that the integration domain I = (0, T ) is used in the definitions of all functionals (1.3), (1.4), and (1.5). Because positivity of the second variation of Λc is only proved for c = 2 independently of the parameter E , see Proposition 1.5, we assume henceforth that c = 2. Lemma 6.2. Assume that ψ is given by (6.4) for some (ξ, θ ) ∈ R2 and some real-valued functions 2 (0, T ) satisfying the orthogonality conditions (6.2). There exist positive constants u, v ∈ Hper C1 , C2 , and 1 such that, if u + iv Hper 2 ≤ 1 , then C1 u + iv 2H 2 ≤ Λc=2 (ψ) − Λc=2 (u0 ) ≤ C2 u + iv 2H 2 . per

per

(6.5)

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Proof. We first note that the functional Λc is invariant under translations and phase rotations in 2 (0, T ), so that Λ (ψ) = Λ (u + u + iv) if ψ satisfies the representation (6.4). Therefore, Hper c c 0 recalling that u0 is a critical point of Λc and using the same notations as in Section 2, we find     Λc (ψ) − Λc (u0 ) = K+ (c)u, u L2 + K− (c)v, v L2 + Nc (u, v), per

per

(6.6)

where Nc (u, v) collects all terms that are at least cubic in (u, v). In particular, there exists a constant C > 0 such that, if u + iv Hper 2 ≤ 1 , we have the estimate   Nc (u, v) ≤ C u + iv 3 2 . H

(6.7)

per

The upper bound in (6.5) holds from the expressions (2.1) and (2.2) for the quadratic part, the estimate (6.7) for the cubic and quartic parts, and the decomposition (6.6). To bound the expression (6.6) from below, we use the spectral properties of the operators K± (c) established in Sections 2, 4, and 5. For periodic waves of small amplitude and for c in the interval (c− , c+ ), we know from Propositions 1.1 and 2.2 that the spectrum of K± (c) in L2 (R) is the union of the nonnegative Floquet–Bloch spectral bands. If K± (c) are considered as operators in L2per (0, T ) with T = 2NT0 , the same result holds except that the Floquet parameter only takes discrete values. In view of the bounds (2.14), this discretization of the Floquet–Bloch spectral bands implies that both K+ (c) and K− (c) have exactly one zero eigenvalue, and that the rest of the spectrum is positive and bounded away from zero. As was already observed, the kernels of K± (c) are due to the symmetries of the NLS equation, and we have the explicit formulas (2.6) for the eigenvectors. Thus, the orthogonality conditions (6.2) mean precisely that u is orthogonal in L2per (0, T ) to the kernel of K+ (c) and v to the kernel of K− (c). Although the results of Propositions 1.1 and 2.2 hold for periodic waves of small amplitude where E is close to one, Proposition 1.5 implies that the same result holds for periodic waves of arbitrary amplitude independently of the parameter E ∈ (0, 1) in the case c = 2. It then follows that there is a positive constant C such that 

 K+ (2)u, u L2 ≥ C u 2L2 per

per

  K− (2)v, v L2 ≥ C v 2L2 .

and

per

per

Using in addition Gårding’s inequality for the elliptic operators K±(c) we conclude that 

 K+ (2)u, u L2 ≥ C u 2H 2 , per

per



K− (2)v, v

 L2per

≥ C v 2H 2 , per

(6.8)

with a possibly smaller constant C. The lower bound in (6.5) is a direct consequence of (6.6), (6.7), and (6.8). 2 Without loss of generality, we assume from now on that C0 0 ≤ 1 , where C0 , 0 , and 1 are 2 (0, T ) is as in the previous lemmas. It then follows from Lemmas 6.1 and 6.2 that, if ψ ∈ Hper close to the orbit of u0 in the sense of the bound (6.3), then C1 d 2 ≤ Λc=2 (ψ) − Λc=2 (u0 ) ≤ C2 C02 d 2 .

(6.9)

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With this estimate at hand, it is now easy to prove that the decomposition (6.1) with the orthogonality conditions (6.2) holds for all t ∈ R if ψ(·, t) is the solution of the cubic NLS equation 2 (0, T ) satisfying the initial bound (1.14), where δ > 0 is small (1.1) with initial data ψ0 ∈ Hper enough so that C0 (C2 /C1 )1/2 δ < 0 .

(6.10)

2 (0, T ) from ψ(·, t) to the orbit of u , in the sense of (6.3). Indeed, let d(t) be the distance in Hper 0 Initially we have d(0) ≤ δ < 0 by (1.14) and (6.10). Let J ⊂ R be the largest time interval containing the origin such that d(t) ≤ 0 for all t ∈ J . As d(t) is a continuous function of time, it is clear that J is closed. On the other hand, for any t ∈ J , we have by (6.9)

  C1 d(t)2 ≤ Λc=2 ψ(·, t) − Λc=2 (u0 ) = Λc=2 (ψ0 ) − Λc=2 (u0 ) ≤ C2 C02 δ 2 , where we have used the crucial fact that Λc is conserved under the evolution defined by the 2 (0, T ). Thus d(t) ≤ C (C /C )1/2 δ <  , hence by continuity cubic NLS equation (1.1) in Hper 0 2 1 0 the interval J contains a neighborhood of t . So J is open, hence finally J = R. This shows that 2 (0, T ) the decomposition (6.1) holds for all t ∈ R with real-valued functions u(·, t), v(·, t) ∈ Hper satisfying the orthogonality conditions (6.2) as well as the uniform bound

u(·, t) + iv(·, t)

2 Hper

≤ C0 d(t) ≤ C02 (C2 /C1 )1/2 δ,

t ∈ R.

This yields the bound (1.15) with  = C02 (C2 /C1 )1/2 δ. To conclude the proof of Theorem 1.8, it remains to show that the modulation parameters ξ and θ are continuously differentiable functions of time t and satisfy the bound (1.16). Lemma 6.3. Assume that the solution ψ(·, t) of the cubic NLS equation (1.1) satisfies d(t) ≤  ≤ 1 for all t ∈ R, where d(t) denotes as in (6.3) the distance to the orbit of u0 . Then the modulation parameters ξ(t), θ (t) given by Lemma 6.1 are continuously differentiable functions of t satisfying (1.16). 2 (0, T )), the proof of Lemma 6.1 shows that ξ(t) and θ (t) depend conProof. As ψ ∈ C(R, Hper tinuously on t. To prove differentiability, we first consider more regular solutions with initial data 4 (0, T ), and then recover the general case by a density argument. For regular solutions, ψ0 ∈ Hper we can differentiate both sides of the decomposition (6.1) and use the cubic NLS equation (1.1) to obtain the evolution system



    ˙ + 2u0 u + u2 + v 2 v, ut = L− v + ξ˙ u0 + ux − θv   −vt = L+ u − ξ˙ vx − θ˙ (u0 + u) + 3u0 u + u2 + v 2 u + u0 v 2 ,

where the operators L± are defined in (2.3). Using the orthogonality conditions (6.2), we eliminate the time derivatives ut , vt by taking the scalar product of the first line with u0 and of the second line with u0 . This gives the following linear system for the derivatives ξ˙ and θ˙ :    ˙   u , L v u0 , (2u0 u + u2 + v 2 )v L2per − L2per ξ 0 B + = u0 , L+ u L2 θ˙ u0 , (3u0 u + u2 + v 2 )u + u0 v 2 per

L2per

 ,

(6.11)

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where B=

 − u 2

0 L2per

0



0 u0 2L2

 +

per

−u0 , ux L2per

u0 , v L2per

u0 , vx L2per

u0 , u L2per

 .

(6.12)

Since u(·, t) + iv(·, t) Hper 2 ≤ C0 d(t) ≤ C0  for all t ∈ R, the second term in the right-hand side of (6.12) is of size O(), hence the matrix B is invertible if  is small enough. Inverting B in (6.11), we obtain a formula for the derivatives ξ˙ , θ˙ where the right-hand side is a continuous 2 (0, T )). By a classical density function of time under the mere assumption that ψ ∈ C(R, Hper argument, we conclude that ξ , θ are differentiable in the general case, and that their derivatives are given by (6.11). Finally, the first term in the right-hand side of (6.11) is of size O(), whereas the second term is O( 2 ), hence |ξ˙ (t)| + |θ˙ (t)| ≤ C for all t ∈ R, where the positive constant C is independent of t . 2 Acknowledgments The authors thank B. Deconinck for pointing out to his work [4] and for helping to compare our analytic formula (1.13) with the results of [4]. The authors also thank M. Haragus for pointing to the intertwining relation (5.3), which helped us to extend the result to periodic waves of large amplitudes and to prove the spectral stability of periodic waves. D.P. is supported by the Chaire d’excellence ENSL/UJF. He thanks members of Institut Fourier, Université de Grenoble for hospitality during his visit (January–June, 2014). Appendix A. Explicit expressions involving Jacobi elliptic functions In this appendix, we derive explicit formulas the generalized eigenvectors of the linearized operators in (5.2) by using Jacobi elliptic functions. In particular, we show how to compute the explicit expression (3.4). Fix E ∈ (0, 1) and let k ∈ (0, 1) be given by (1.13). The periodic wave profile u0 defined in (1.7) can be rewritten in the explicit form  u0 (x) =

  2k 2 x sn √ ,k = 1 + k2 1 + k2



  2k 2 x j √ , 1 + k2 1 + k2

x ∈ R,

where j (ξ ) = sn(ξ, k) denotes the Jacobi elliptic √ function. To simplify the calculations below, it is convenient to use the space variable ξ = x/ 1 + k 2 instead of x. Let us recall a few properties of the Jacobi elliptic functions sn(ξ, k), cn(ξ, k), and dn(ξ, k) [10]. The functions sn(ξ, k) and cn(ξ, k) are periodic with period T = 4K(k), where K(k) denotes the complete elliptic integral of the first kind. On the other hand, the function dn(ξ, k) = 1 − k 2 sn(ξ, k)2 is periodic with period 2K(k). We have the following expressions for the first-order derivatives of the Jacobi elliptic functions: ⎤ ⎡ ⎤ ⎡ sn(ξ, k) cn(ξ, k) dn(ξ, k), d ⎣ cn(ξ, k) ⎦ = ⎣ − sn(ξ, k) dn(ξ, k), ⎦ dξ dn(ξ, k) −k 2 sn(ξ, k) cn(ξ, k).

(A.1)

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In particular, the function j (ξ ) = sn(ξ, k) satisfies the differential equation   d 2j = − 1 + k 2 j + 2k 2 j 3 . dξ 2

(A.2)

Let us also introduce the incomplete elliptic integral of the second kind ξ E(ξ, k) =

dn2 (y, k) dy,

ξ ∈ R.

(A.3)

0

This function is not periodic and we have the relation   E ξ + 2K(k), k = E(ξ, k) + 2E(k) for all ξ ∈ R, where E(k) := 12 E(2K(k), k) is the complete elliptic integral of the second kind. This means that the function ξ → E(ξ, k) is linearly growing at infinity with asymptotic rate E(k)/K(k). Using the chain rule for the operator L− = −∂x2 + u20 (x) − 1, we obtain L− = (1 + k 2 )L− , where   L− = −∂ξ2 − 1 + k 2 + 2k 2 j (ξ )2 . Recall that L− j = 0. Using the relations (A.1)–(A.3), it is easy to verify that   L− cn(ξ, k) dn(ξ, k) = −4k 2 cn(ξ, k) dn(ξ, k) sn2 (ξ, k),     L− sn(ξ, k)E(ξ, k) = −2 cn(ξ, k) dn(ξ, k) 1 − 2k 2 sn2 (ξ, k) ,   L− ξ sn(ξ, k) = −2 cn(ξ, k) dn(ξ, k). Therefore, the function   E(k) V (ξ ) := cn(ξ, k) dn(ξ, k) + sn(ξ, k) E(ξ, k) − ξ , K(k) is periodic with period T = 4K(k) and satisfies the inhomogeneous equation     E(k) E(k)  L− V = −2 1 − cn(ξ, k) dn(ξ, k) = −2 1 − j. K(k) K(k)

(A.4)

(A.5)

Note that the numerical coefficient in (A.5) is nonzero because K(k) > E(k) for all k ∈ (0, 1). Using the chain rule for the operator M− = ∂x4 − 3∂x u20 ∂x + u20 − 1, we obtain M− = (1 + k 2 )2 M− , where   2  M− = ∂ξ4 − 6k 2 ∂ξ j (ξ )2 ∂ξ + 2k 2 1 + k 2 j (ξ )2 − 1 + k 2 . A long but direct calculation using (A.1) shows that the same function V in (A.4) also satisfies      E(k)  M− V = 4 k 2 − 1 − 1 + k2 j . K(k)

(A.6)

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Recall that K− (c) = M− − cL− . Combining (A.5) and (A.6) and using the chain rule, we obtain         E(k) M− − c 1 + k 2 L− V = 4k 2 + 2(c − 2) 1 + k 2 1 − j . K(k)



(A.7)

Note that j and V are orthogonal with respect to the scalar product · ,· L2per in L2per (−2K(k), 2K(k)) because V is even and j is odd. Remark A.1. The fact that both quantities L− V and M− V are proportional to the same function j  is not an accident. Associated with the neutral mode (u0 , 0), we have L− v = u0 arising in the solutions of the linearized evolution operator at u0 : 

0 −L+

L− 0



   u0 0 = 0 0

 and

0 −L+

L− 0

     0 u = 0 , v 0

hence (0, v) is the generalized neutral mode. Now the higher-order operators M± are associated with the linearization of another flow in the hierarchy of the integrable NLS equation, which commutes with the original flow of (1.1), see Section 5. As is easily verified, this implies that the same function v satisfies M− v = Au0 for some constant A ∈ R, in agreement with (A.5) and (A.6) after the scaling transformation from x to ξ . We can now obtain the explicit expression (3.4) from the formula (3.1). Recall that z = x and U = u0 (−1 ·). If W = w(−1 ·) satisfies P− (c, 0)W = U  , then 

−2       P− (c, 0)W (z) = 1 + k 2 M− − c 1 + k 2 L− w(ξ ),

z ξ= √ .  1 + k2

Using the chain rule, we rewrite the formula (3.1) in the equivalent form μ (0) =

    −1   22 −4(c − 2)2 1 + k 2 j  , M− − c 1 + k 2 L− j  L2 2 per j L2 per

−1 2

 + 3 1 + k 2 j  L2 + (3 − c) j 2L2 . per

(A.8)

per

It follows from (A.7) that 

 −1  M− − c 1 + k 2 L− j  =

V 4k 2

+ 2(c − 2)(1 + k 2 )(1 −

E(k) K(k) )

.

(A.9)

It remains to compute the norms and the scalar products in the right-hand side of Eq. (A.8). Using the notations above, we find for all k ∈ (0, 1), j 2L2 per

 2

j 2 L

per

  4K(k) E(k) = > 0, 1− K(k) k2    2  E(k) 4K(k) 2 = − 1 + k + 1 > 0, k K(k) 3k 2

(A.10) (A.11)

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and 



j ,V

 L2per

  2K(k) 2 E(k) E(k)2 = − k −1+2 . K(k) K(k)2 k2

(A.12)

Substituting these expressions into (A.8) and finding a common denominator for all terms, we obtain the expression (3.4). Next, using the chain rule for the operator L+ = −∂x2 + 3u20 (x) − 1, we obtain L+ = (1 + k 2 )L+ , where   L+ = −∂ξ2 − 1 + k 2 + 6k 2 j (ξ )2 . Recall that L+ j  = 0. Using the relations (A.1)–(A.3), it is easy to verify that   L+ sn(ξ, k) = 4k 2 sn3 (ξ, k), 

ξ

L+ cn(ξ, k) dn(ξ, k)

sn2 (y, k) 2

0

dn (y, k)



  dy = −2 sn(ξ, k) 1 − 2 sn2 (ξ, k) ,

    L+ ξ cn(ξ, k) dn(ξ, k) = 2 sn(ξ, k) 1 + k 2 − 2k 2 sn2 (ξ, k) . Therefore, the function    U (ξ ) := 1 − k 2 1 + bk 2 sn(ξ, k) 

−k 1−k 2

2



ξ cn(ξ, k) dn(ξ, k) 0

sn2 (y, k) dn2 (y, k)

 dy − bξ ,

(A.13)

satisfies the inhomogeneous equation         L+ U = 2k 2 1 − k 2 1 + b 1 + k 2 sn(ξ, k) = 2k 2 1 − k 2 1 + b 1 + k 2 j,

(A.14)

for an arbitrary coefficient b ∈ R. We shall find the value of b from the condition that U is periodic with period T = 4K(k). To do so, we recall the identity (see 16.26.6 in [1]): 

1−k

2



ξ

dy 2

0

dn (y, k)

= E(ξ, k) − k 2

sn(ξ, k) cn(ξ, k) . dn(ξ, k)

Using this identity, we rewrite the function U given by (A.13) in the equivalent form    U (ξ ) = 1 − k 2 1 + bk 2 sn(ξ, k) + k 2 sn(ξ, k) cn2 (ξ, k)   

− cn(ξ, k) dn(ξ, k) E(ξ, k) − 1 − k 2 1 + bk 2 ξ ,

(A.15)

T. Gallay, D. Pelinovsky / J. Differential Equations 258 (2015) 3607–3638

which is periodic if and only if (1 − k 2 )(1 + bk 2 ) = we finally obtain the 4K(k)-periodic solution U (ξ ) =

3637

E(k) K(k) . Substituting this expression into (A.15),

E(k) sn(ξ, k) + k 2 sn(ξ, k) cn2 (ξ, k) K(k)   E(k) − cn(ξ, k) dn(ξ, k) E(ξ, k) − ξ K(k)

(A.16)

of the inhomogeneous equation     2 2 E(k) L+ U = 2 k − 1 + 1 + k j. K(k)

(A.17)

Note that the numerical coefficient in (A.17) is nonzero for every k ∈ (0, 1), thanks to (A.11). Using the chain rule for the operator M+ = ∂x4 − 5∂x u20 ∂x − 5u40 + 15u20 − 4 + 3E 2 , we obtain M+ = (1 + k 2 )2 M+ , where     M+ = ∂ξ4 − 10k 2 ∂ξ j (ξ )2 ∂ξ − 20k 4 j (ξ )4 + 30k 2 1 + k 2 j (ξ )2 − 1 + 14k 2 + k 4 . After a long but direct calculation, we obtain that the same function U in (A.16) also satisfies    E(k)  M+ U = 4 2k 4 − k 2 − 1 + 1 + 4k 2 + k 4 j. K(k)

(A.18)

Combining (A.17) and (A.18) into K+ (c) = M+ − cL+ for c = 2 and using the chain rule, we obtain       2E(k) 2 2 2 M+ − 2 1 + k L+ U = 4k k − 1 + j. (A.19) K(k) Since 2 > 1 + k 2 , the numerical coefficient in front of j is positive for all k ∈ (0, 1), thanks to (A.11). Remark A.2. Again, we observe that both quantities L+ U and M+ U are proportional to the same function j . This is due to the generalized neutral mode (u, 0) associated with the neutral mode (0, u0 ), which arise in the solution of L+ u = u0 . See also Remark A.1. References [1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, NY, 1972. [2] M.A. Alejo, C. Munoz, Nonlinear stability of MKdV breathers, Comm. Math. Phys. 324 (2013) 233–262. [3] J. Angulo Pava, Nonlinear Dispersive Equations. Existence and Stability of Solitary and Periodic Travelling Wave Solutions, Math. Surveys Monogr., vol. 156, Amer. Math. Soc., Providence, RI, 2009. [4] N. Bottman, B. Deconinck, M. Nivala, Elliptic solutions of the defocusing NLS equation are stable, J. Phys. A 44 (2011) 285201, 24 pp. [5] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Amer. Math. Soc. Colloq. Publ., vol. 46, Amer. Math. Soc., Providence, RI, 1999.

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